A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

1, -1, 1, 5, -21, -399, 869, 39325, -334477, -28717403, 59697183, 8400372435, -34429291905, -7199255611995, 14631594576045, 4251206967062925, -68787420596367165, -26475975382085110035, 53392138323683746235, 26275374869163335461975, -105772979046693606062363

Offset

1,4

Links

Table of n, a(n) for n=1..21.

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. [Except for the signs, see the unnumbered table on p. 7.]

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. [Except for the signs, see Table 4.]

D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.

Eric Weisstein's World of Mathematics, Gamma Function.

Example

1/sqrt(x^(-1)) - sqrt(x^(-1))/8 + (x^(-1))^(3/2)/128 + (5*(x^(-1))^(5/2))/1024 - (21*(x^(-1))^(7/2))/32768 + ...

Maple

H := proc(n) local S, i; S := (x/(exp(x)-1))^(3/2)*exp(x/2);
-pochhammer(1/2, n-1)*coeff(series(S, x, n+2), x, n)*2^(4*n-1-add(i, i= convert(n, base, 2))) end:
A143503 := n -> (-1)^irem(n-1, 6)*H(n-1);
seq(A143503(n), n=1..16); # Peter Luschny, Apr 05 2014

Mathematica

Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* Vaclav Kotesovec, Oct 09 2023 *)

Crossrefs

Cf. A061549, A088802 (denominators), A222411, A222412.

Sequence in context: A006109 A009732 A009758 * A144779 A193324 A220002

Adjacent sequences: A143500 A143501 A143502 * A143504 A143505 A143506

Keyword

sign,frac

Author

Eric W. Weisstein, Aug 20 2008

Extensions

More terms from Vaclav Kotesovec, Oct 09 2023

Status

approved